The German fleet and its diplomatic importance is probably the most misunderstood event of the era.
First, the Germans had a host of grievances against the British. Beginning with Roseberry and continuing with the last Salisbury administration, Britain repeatedly undercut the Triple Alliance. There are many but a short list: the Congo treaty which would have cut Germany off from the Free State, support for the Armenian rebels, the refusal to support the Triple Intervention and, mot importantly, the lack of support for Italy in Ethiopia. The latter leads to the downfall of Crispi, the end of the Franco-Italian rivalry and the end of the effective end of the Triple Alliance. The Germans, correctly, saw British policy as to create tensions between the Triple Alliance and France/Russia which were followed by Britain selling out the Triple Alliance
Chamberlain's so called "alliance" offers to Germany should be seen in this light. Each were a response to Russia in China (where Britain had interests and Germany didn't). Germany was supposed to risk Russian wrath to support British policy while Britain offered little real support to Germany. Naturally, the Germans wanted nothing to do with this.
Second, the Entente was conceived long before the "German threat" emerged. Negotiations had begun seriously after the Anglo-Japanese Alliance was formed as a check on Russia in Asia and before the Russo-Japanese War. Rather than a response to the German fleet, it was an effort to detach France from Russia. The Franco-Russian Alliance was the real threat to British interests in 1903. The idea that the British sought an Entente with the second and third largest navies because she feared the 5th larges one is preposterous. Instead, Britain was settling her disputes with France at the expense of Germany just like she had so many times before. The Kaiser was right: Germany's treaty rights in Morocco were being ignored.
Third, the German fleet expansion corresponds to a period of warm relations with Russia. Between the ascension of Nicholas (1894) and the Bosnian crisis (1908), there were no major issues between Russia and the Triple Alliance. The 1897 Austro-Russian freezing of Balkan issues being the prime example. The relations were so good, that Germany would offer Russia an Alliance proposal. The issues of the day were overseas (mainly China and Africa) and so having a strong navy that would lead both Britain and France/Russia bid for German support made perfect sense.
Germany had her own interests. The Kaiser's job was to protect them not Britain's. Unfortunately, too many textbooks are written by the victors and we get the British view more than the German one.
Ah yes too many textbooks are written by the other side, how dare they.
There are a few flaws with the narrative presented and the key ones are dates. The First German Naval Law was enacted in 1898 and the Second in 1900 both were part of the Risk strategy aimed at Britain, the whole the British were reacting before the Germans tried anything in 1903 is a bit backwards, rather consciously so. Not that the narrative of some kind of British duty to a Triple Alliance does not sound odd.
Germany and in particular the Kaiser himself would manipulate the Russian Tsar into war with Japan without any intention of providing actual support, so I suppose his apologists projecting his motives onto Britain makes sense. Which is not to say Britain was a good guy in all of this, it was a bully but it was a bully because it had the muscle to back it up, financial and naval and in the colonial sphere an ability to project power on land too. Thus Tirpitz's idea might have made some sense if Germany had the means to enable.
Certainly there was a sense that other powers would bid for a German alliance. This was a fairly typical over estimation of Germany's own capacities and led to the failure of pretty much all its alliance proposals with either Britain or America or indeed any powers not in its near abroad.
Now as to the size of the German Navy just prior to Dreadnought they had 25 battleships, as many as France, the USN or (if we count 4 coastal defence battleships) Russia. The British had depending on which ones exactly are counted 53-57, some were obviously getting rather old at this point but 2 more pre-Dreadnoughts of the Lord Nelson class were under construction. Now it may be pointed out that a number of the German battleships were rather weak with none having guns of larger than 11' calibre and some having 9.4' guns, the Germans had tried with the Brandenburg class to field battleships with 6 main guns, the then normal pair fore and aft being joined with a shorter pair in an amidships turret. Mind you in German battleships were rather weak then some of the French ones were just plain odd so combat power was perhaps comparable.
The thing was that the British never had nor intended a fleet able to take all comers. They simply intended a fleet able to take on one power while preventing another power jumping on their back. Nor was the size of a given navy a particular issue, the British could and would remain able for some time to outbuild any one navy. It was the intent. That Germany had issues with British policy was understood, so too did the French, Russians, Americans, Italians, Japanese (yes and they were allies) and so on. The idea of someone building a navy to protect from the Royal Navy was, as I have pointed out above, neither new nor controversial.
That however was not what Tirptiz's risk theory was all about. It was very pointedly about designing a force to extort concessions from the British. Hence the focused paranoia. The clear and documented timing also matches with Britain's increased focus on mollifying other potential rivals to concentrate against its one avowed enemy and her allies. The problem was that Germany's naval-industrial complex was not even close to being able to match the supplier of one third of the world's warships and a rather greater proportion of its merchant shipping. In addition the finances of the German Empire were always more precarious than those of the United Kingdom (even without being able to further leverage its Imperial Dominions).
Hence the view then, and that was important, in chancelleries across Europe and the Americas that Germany was on collision course with Britain. An understanding that was leveraged to their own advantage by France and Russia and others.
That the British were selfish thugs is a valid point but there is no getting around that the problem the Germans had was their leaders, however gifted in certain spheres, were idiots in how they handled the wider implications of strategic diplomacy.